Momentum Operator in One Dimension
In quantum mechanics, physical observables like momentum are represented by mathematical operators. These operators act on the system's wavefunction to provide information about the observable.
For a particle moving in one dimension (along the x-axis), the momentum operator, often denoted as $\hat{p}_x$, is defined as:
$ \hat{p}_x = -i\hbar \frac{d}{dx} $
Where:
- $i$ is the imaginary unit ($\sqrt{-1}$).
- $\hbar$ is the reduced Planck constant ($h / (2\pi)$).
- $\frac{d}{dx}$ is the differential operator representing the derivative with respect to position $x$.
This operator is fundamental in the formulation of the Schrödinger equation and is used to calculate the momentum expectation value and related properties of a quantum system.
Comparing this definition with the options provided:
- Option 1: $i\hbar \frac{d}{dx}$ is incorrect.
- Option 2: $-i\hbar \frac{d}{dx}$ matches the definition.
- Option 3: $i\hbar \frac{\partial}{\partial t}$ is the energy operator (Hamiltonian) in time-dependent form, not momentum.
- Option 4: $-\frac{\hbar^2}{2m} \frac{d^2}{dx^2}$ is the kinetic energy operator.
Therefore, the correct quantum mechanical operator for the momentum of a particle moving in one dimension is $-i\hbar \frac{d}{dx}$.
Important Questions from Operators Commutators Heisenberg Picture
- Consider an operator $\hat{A}$ which is not Hermitian. Find the possible values of $c$ and $d$ such that the operator $(c\hat{A} - d\hat{A}^\dagger)$ is Hermitian.
- Which of the following operators is/are self-adjoint?
- Consider operators $\hat{A}$, $\hat{B}$, and $\hat{C}$ for three observables of a quantum system satisfying $[\hat{A}, \hat{B}] = 0$, $[\hat{B}, \hat{C}] = 0$, and $[\hat{A}, \hat{C}] \neq 0$, with uncertainties $\Delta A, \Delta B, \Delta C$, respectively. From the options given below, which is/are implied by the commutation relations among $\hat{A}, \hat{B}$, and $\hat{C}$?
- Let $|m\rangle$ and $|n\rangle$ denote the energy eigenstates of a one-dimensional simple harmonic oscillator. The position and momentum operators are $\hat{X}$ and $\hat{P}$, respectively. The matrix element $\langle m|\hat{P}\hat{X}|n\rangle$ is non-zero when
The wavefunction of a particle in one dimension is given by
$\psi(x) = \begin{cases} M, & -a < x < a \\ 0, & \text{otherwise.} \end{cases}$
Here $M$ and $a$ are positive constants. If $\phi(p)$ is the corresponding momentum space wavefunction, which one of the following plots best represents $|\phi(p)|^2$ ?
